# Convergence/Divergence of Integrals

• amcavoy
In summary, the conversation discussed determining whether the integral \int_{0}^{1-}\frac{dx}{\sqrt{1-x^4}} converges or diverges without evaluating it. The function behaves similarly to 1/x^2, but it is not integrable over the domain [0,1]. It was suggested to find a function that is greater than the given function and has a finite integral on [0,1]. The integral is evaluated in terms of the Gamma function, specifically the Legendre elliptic integral, and expressed in terms of the Gamma factor. A substitution can transform the integral to \int_{1}^{+\infty} \frac{dx}{\sqrt{x^{4}-1}},
amcavoy
Determine whether the following integral converges or diverges without evaluating it.

$$\int_{0}^{1-}\frac{dx}{\sqrt{1-x^4}}$$

That function behaves ~1/x^{2},so it should be integrable in that domain.U have to find a function which is greater than your function,however,it's integral on [0,1] needs to be finite...

$$\int_{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} =\frac{1}{4}\left(\sqrt{\pi}\right)^{3}\frac{\sqrt{2}}{\Gamma^{2}\left(\frac{3}{4}\right)}$$

Daniel.

The problem point is x=1, not x=0. √(1 - x4) behaves like 2√(1-x) there.

P.S. 1/x2 isn't integrable over [0, 1].

I didn't say "integrable".And i didn't say =1/x^{2}...

Daniel.

i believe it certainly converges. now let me think why i say that.

if y^2 = 1-x^4 then 2ydy = -4x^3 dx, so dx/sqrt(1-x^4) = dx/y = -2dy/4x^3. which makes perfectly good sense at x = 1.

huh?

formulas have always seemed like magic to me.

Last edited:
Yeah,but it's still infinite,because your limits of integration will still involve 0...

Daniel.

dextercioby said:
That function behaves ~1/x^{2},so it should be integrable in that domain.U have to find a function which is greater than your function,however,it's integral on [0,1] needs to be finite...

$$\int_{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} =\frac{1}{4}\left(\sqrt{\pi}\right)^{3}\frac{\sqrt{2}}{\Gamma^{2}\left(\frac{3}{4}\right)}$$

Daniel.

Can you explain how this integral is evaluated in terms of the Gamma function?

I dunno.Basically,it returned my a value for the Legendre elliptic integral and then expressed this one in terms of Gamma factor...

Daniel.

If u make a substitution,u can transform the integral to

$$\int_{1}^{+\infty} \frac{dx}{\sqrt{x^{4}-1}}$$

whose antiderivative,Mathematica finds

Daniel.

P.S.My Maple gave me the results...

#### Attachments

• Integrate.gif
1.2 KB · Views: 546

## What is the definition of convergence and divergence of integrals?

The convergence or divergence of integrals is a concept in mathematics that determines whether a given integral, or the area under a curve, will approach a finite value (converge) or an infinite value (diverge) as the limits of integration approach certain values.

## How do you determine if an integral converges or diverges?

To determine the convergence or divergence of an integral, you must evaluate the integral using the limit comparison test or the comparison test. These tests compare the given integral to a known convergent or divergent integral and determine if they have similar behavior.

## What is the difference between absolute and conditional convergence?

Absolute convergence is when the integral of a function converges regardless of the order in which the terms are summed. Conditional convergence is when the order of summation affects the convergence of the integral.

## What is the role of the limit in determining convergence or divergence of integrals?

The limit is crucial in determining the convergence or divergence of integrals because it represents the behavior of the integral as the limits of integration approach certain values. If the limit approaches a finite number, the integral will converge, but if the limit approaches infinity, the integral will diverge.

## Can improper integrals converge or diverge at both limits?

Yes, improper integrals can converge or diverge at both limits. This is known as a bi-directional improper integral. In this case, the integral will converge if both limits approach finite values or if one limit approaches a finite value while the other approaches infinity. The integral will diverge if both limits approach infinity or if one limit approaches a finite value while the other approaches negative infinity.

Replies
15
Views
3K
Replies
4
Views
1K
Replies
3
Views
1K
Replies
29
Views
2K
Replies
2
Views
858
Replies
4
Views
1K
Replies
12
Views
2K
Replies
9
Views
944
Replies
3
Views
284
Replies
11
Views
2K